Question Video: Calculating a Correlation Coefficient from Calculated Sum of Squares

A data set has summary statistics 𝑆_(π‘₯π‘₯) = 36.875, 𝑆_(𝑦𝑦) = 73.875, and 𝑆_(π‘₯𝑦) = 32.375. Calculate the product-moment correlation coefficient for this data set, giving your answer correct to three decimal places.

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Video Transcript

A data set has summary statistics 𝑆 π‘₯π‘₯ is equal to 36.875, 𝑆 𝑦𝑦 is 73.875, and 𝑆 π‘₯𝑦 is 32.375. Calculate the product-moment correlation coefficient for this data set, giving your answer correct to three decimal places.

We are given the summary statistics for a data set, where 𝑆 π‘₯π‘₯ is 36.875; that’s the variation in π‘₯. 𝑆 𝑦𝑦 is 73.875; that’s the variation in 𝑦. And 𝑆 π‘₯𝑦 is 32.375; that’s the covariance of π‘₯ and 𝑦. And using these summary statistics, we want to calculate the product-moment correlation coefficient. To do this, we can use the abbreviated form of the correlation coefficient formula. That is, the correlation coefficient π‘Ÿ π‘₯𝑦 is 𝑆 π‘₯𝑦 divided by the square root of 𝑆 π‘₯π‘₯ times 𝑆 𝑦𝑦.

Since we’re given the summary statistics, we simply need to substitute these into the formula. So that π‘Ÿ π‘₯𝑦 is 32.375, that’s 𝑆 π‘₯𝑦, divided by the square root of the product of 36.875, which is 𝑆 π‘₯π‘₯, and 73.875, which is 𝑆 𝑦𝑦. That is 32.375 over the square root of 2724.140625, which is 32.375 divided by 52.19330 to five decimal places. And that’s approximately 0.62029. And so to three decimal places, the product-moment correlation coefficient for this data set is π‘Ÿ π‘₯𝑦 is equal to 0.620.

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